TITLE: Lower regularity version of Moser's theorem on volume elements
QUESTION [6 upvotes]: A theorem of Moser, published in "On the Volume Elements of a Manifold" (Transactions of the Americal Mathematical Society 120, 1965; doi: 10.1090/S0002-9947-1965-0182927-5, jstor), shows that if a $C^\infty$ compact manifold $M$ has two $C^\infty$ volume forms $\omega_1$ and $\omega_2$ with the same total mass, then there is a diffeomorphism of $M$ sending one to the other. 
I am interested in what is known if the manifold and volume forms have lower regularity (in particular, I really want to know about the $C^{1+\alpha}$ case. 
Thanks for any reference suggestions.
EDITED:
So having had an answer from Robert Bryant, I realized I should have been more precise about the specific question(s) that I was asking:

If $M$ is a $C^{1+\alpha}$ manifold and $\omega_i$, $i=1,2$ are two $C^\alpha$
volume forms with the same mass, does there exist a $C^{1+\alpha}$ diffeomorphism sending one the other?

The comment below from AlvarezPaiva suggests the answer to the above might be yes, but the context there appears to be bounded subsets of $\mathbb R^n$.

If $M$ is a $C^{1}$ manifold and $\omega_i$, $i=1,2$ are two continuous
volume forms with the same mass, does there exist a $C^1$ diffeomorphism sending one to the other?

Final full disclosure in case this drastically simplifies things: my manifold is topologically a two-dimensional torus.

REPLY [11 votes]: If $M$ is a $C^{1}$ manifold and $\omega_i$, $i=1,2$ are two continuous
volume forms with the same mass, does there exist a $C^1$ diffeomorphism sending one to the other?

The answer is no. The equivalent problem is: given a positive and continuous function $g$, can we find a diffeomorphism with the Jacobian equal to $g$, $\det Df=g$? A counterexample is Theorem 1.2  in [1].

If $M$ is a $C^{1,\alpha}$ manifold and $\omega_i$, $i=1,2$ are two $C^\alpha$
volume forms with the same mass, does there exist a $C^{1,\alpha}$ diffeomorphism sending one the other?

The answer is yes, at least locally (see also a comment of  alvarezpaiva). Whether the result has a global version on manifolds, I do not know. This is Theorem 1 in [3]. For a comprehensive treatment of related results, see [2].

Theorem. Let $k\geq 0$ and $\alpha\in (0,1)$. If $\Omega\subset\mathbb{R}^n$ is abounded domain with $C^{k+3,\alpha}$
  boundary and $\omega\in C^{k,\alpha}$ is a volume form such that
  $\int_\Omega\omega=|\Omega|$, then there is a diffeomorphism
  $\varphi:\Omega\to\Omega$ that is identity on the boundary,
  $\varphi,\varphi^{-1}\in C^{k+1,\alpha}(\bar{\Omega})$ and such that
  $\varphi^*\omega=dx_1\wedge\ldots\wedge dx_n$.

[1] D. Burago, B. Kleiner, Separated nets in Euclidean space and Jacobians of bi-Lipschitz maps.
Geom. Funct. Anal. 8 (1998), 273–282. 
(MathSciNet review.)
[2] G. Csató, B. Dacorogna, O. Kneuss, The pullback equation for differential forms. Progress in Nonlinear Differential Equations and their Applications, 83. Birkhäuser/Springer, New York, 2012. 
(MathSciNet review.)
[3] B. Dacorogna, J. Moser,
On a partial differential equation involving the Jacobian determinant. 
Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990),  1–26. 
(MathSciNet review.)